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Advanced Micro Theory Preferences and Utility Consumer Choice • Postulate: an unproved and indemonstrable statement that should be taken for granted: used as an initial premise or underlying hypothesis in a process of reasoning • Consumer choice postulate: People choose from available options to maximize their wellbeing (utility). Criticisms • Criticisms – What about irrational consumers? – Can consumers make these internal calculations? • Irrelevant. We just want to successfully predict behavior. To do that, we assert that all consumers behave accordingly. • Refutation comes if theorems that derive from this postulate are inconsistent with the data. – That is, if behavior contradicts the implications of the model, then the theory is wrong. Alternatives • We could devise a hypothesis that postulates that consumers – act randomly – do what they think society wants them to do • But all behavior would be consistent with these assumptions, so no refutable implications (theoretical results) are possible… therefore, a theory based on such a hypothesis is useless. Consumer Choice Model • “People choose from available options to maximize their well-being (utility).” – “Available options” in the model will be handled by the budget constraint. • The budget constraint will provide decision-makers with MC of choices. – “Maximizing well-being” will be incorporated into the model via assumptions about preferences – which will then be used to build a utility function. • The preferences part of the model will provide decisionmakers with the MB of choices. Modeling Preferences • Let bundle A = (x1, y1) and B = (x2, y2) where the goods are x and y. Y A y1 y2 B x1 x2 X Varian’s Version • Let bundle X = (x1, x2) and Y = (y1, y2) where the goods are x1 and x2 . So the goods listed on the axes and the quantities of each good in the first bundle are the using the same notation. X2 X x2 y2 Y x1 y1 X1 Varian’s Version • He does this to be consistent with his advanced micro text. good 2 In that text, he uses vector notation and eliminates the subscripts by not noting quantities of each good on the axes. X Y good 1 Modeling Preferences • IMO, students have invested so much math time with X and Y on the axes, dy that I want to leverage that. E.g. slope = dx • Also, with all the derivations coming up, we will have plenty of subscripts floating around that I hate to add an additional set with goods X1 and X2. Y A y1 y2 B x1 x2 X Modeling Preferences • Three choices: A B, consum er strictly prefers bundle A to bundle B A B, consum er w eakly prefers bundle A to bundle B A B, consum er indifferent betw een bundle A and bundle B Y • And therefore A If A B, and B A then A If A B, and not A B then A y1 y2 B x1 x2 B X B Axioms of Preference • Axiom: a proposition that is assumed without proof for the sake of studying the consequences that follow from it (dictionary.com). – These are based on ensuring logical consistency. • Completeness: A B, or B A , or both, m eaning A B. – Any pair of bundles can be compared and ordered • Reflexivity: A A , or A A – A bundle cannot be strictly preferred to an identical bundle. • Transitivity Let C = (x 3 ,y 3 ) If A B, and B C then A C – Not a logical imperative according to Varian, but preferences become intractable if people cannot choose between three bundles because A B, and B C and C A • Continuity, next page Continuity • Preferences must be continuous Rules out this situation: • • • • Y Ub=15 B Ua=10 Uc=20 A The bundle with more X is always preferred. Holding X constant, more Y is better. B A, C A But, no matter how close C gets to A, C B The utility function in this case must be discontinuous (i.e. there must be a vertical jump between A and C C X Goods, Bads and Neutral Goods • • • • Goods are good (more is better) Bads are bad, less is better Neutrals mean nothing to the consumer Some goods start out good, but then become bads if you consume too much Possible Indifference Mappings Thus Far Characterize the Goods Y Y Y X X X X Y Y Y X X And we have… Y Both are good but become bad Y Y is a neutral good Y Y Two bads X X X Y Two goods X good and Y bad Y X good that becomes bad, Y good X X X Perfect Compliments and Substitutes Y Perfect Compliments: More is only better if you have more of the other X Y Perfect Substitutes: Two goods, indifferent to trading off a constant amount of Y for X X Well-behaved Preferences • If we want to avoid situations where demand curves are upward sloping or people spend all their money on one good, then we need wellbehaved indifference curves. • Preferences must also be – Monotonic – Convex Monotonic • Monotonic: If bundle A is identical to B, except A has more of at lease one good, then A B – A.K.A, nonsatiation or “more-is-better” – Ceteris paribus, increasing the quantity of one good creates a bundle that is strictly preferred. – Indifference curves must be downward sloping. – Paired with transitivity, means indifference curves cannot cross. Monotonic • These still possible Y Y Y X X X Indifference Curves Cannot Cross A C , share an indifference curve B C , share an indifference curve A Y A B, transitivity B ut A B C X B, m onotonicity Convexity • Convexity: People prefer more balanced bundles. – Let A = (x1, y1) and B = (x2, y2). – Define C = (tx1 + (1-t)x2, ty1 + (1-t)y2) • where 0 ≤ t ≤ 1 – then C A and C Y B A y1 C t y1 + (1-t)y2 B y2 x1 tx1 + (1-t)x2 x2 X Convexity: Indifference Curves Bound Convex Sets • Convexity: Bundles weakly preferred to those lying on an indifference curve bound a convex set. – Any bundle which is a weighted average of bundles on the indifference curve are weakly preferred to bundles lying on the curve. Y Y A y1 A y1 B y2 y2 x1 x2 B X x1 x2 X Convex Preferences • These still possible Y Y X X Strict Convexity Indifference Curves Bound Convex Sets • Strictly Convex Preferences: – Any bundle which is a weighted average of bundles on the indifference curve are strictly preferred to bundles lying on the curve (weights 0 > t > 1). Y C A, C B A y1 C ty1 + (1-t)y2 B y2 x1 tx1 + (1-t)x2 x2 X – Simple convexity allows for straight line segments of the indifference curve – Strict convexity does not, the curve must have an increasing slope as X increases. Convexity • Intuition: people prefer balanced bundles of goods to bundles with a lot of one good and little of the other good. U=4 Y U=7 Along a straight line connecting the axis, Utilty will rise and then fall. U=10 U=7 U=4 X Convexity: Intuition • Which implies indifference curves bow towards the origin. U=4 Y U=7 U=10 U=7 U=4 X Marginal Rate of Substitution • The change in the consumption of the good on the Y axis necessary to maintain utility if the consumer increases consumption of the good on the X axis by one unit. • MRS = the slope of the indifference curve. dy • Although, dx 0 , MRS is almost always defined as the abs value of the slope. • In this class, M R S dy dx MRS = MB • The MRS describes, at any given point along the indifference curve, the consumer’s willingness to give up Y for one more X. • It is therefore the marginal willingness to pay for X • I.e. it is the marginal benefit of consuming X. Digression: Cardinal Utility • Utilitarians believed that utility, like temperature or height, was something that was measurable (Cardinal utility). – – • Early neoclassical economists (e.g. Marshall) still held the idea that for an individual, utility may be a cardinal measure. – – – • And that a unit of utility was the same for everyone so if we could find out how to measure it, we could redistribute to maximize social welfare. The hope of some way of measuring utils did not survive long. Believed marginal utility was strictly decreasing. Marshall’s demand curve was downward sloping for this reason. He is the reason P is on the vertical axis. Diminishing willingness to pay reflected diminishing marginal utility. Also believed that utility was additive, consumption of one good did not affect the MU from another (Uxy = 0). Digression: Ordinal Utility • Pareto (1906) first considered the idea that ordinal utility (ordering the desirability of different choices) might be the way to think of utility. • Work by Edgeworth, Fischer and Slutsky advanced the theory. • Hicks and Allen (1934) came up with the defining theory… that we still use today. • Pareto, Vilfredo (1906). "Manuale di economia politics, con una introduzione ulla scienza sociale". Societa Editrice Libraria. • Viner, Jacob. (1925a). "The utility concept in value theory and its critics". Journal of political economy Vol. 33, No. 4, pp. 369-387 • Hicks, John and Roy Allen. (1934). "A reconsideration of the theory of value". Economica Vol. 1, No. 1, pp. 52-76 The Utility Function • A utility function is simply a way to mathematically represent preferences. • Utility is Ordinal: The ability to order bundles is all that matters. – The magnitude of the difference in utility is meaningless as the numbers reflecting utility are arbitrary. – No interpersonal comparisons are possible. The Utility Function • A function such that A B if and only if U (A ) U (B ) • Preference can be represented by a continuous U=U(A) so long as preferences are reflexive, complete, transitive, continuous • Note, monotonacity and convexity are not needed. • Monotonacity is always assumed because it makes the existence proof easier and more intuitive. The Utility Function • While we need preferences to be reflexive, complete, transitive, continuous for utility functions to exist. • We need monotonacity and convexity to make them well behaved. • By well behaved, we want unique solutions that are not extreme and are relatively stable. – We don’t want individuals spending all their income on one good or slight changes in price or income to drastically affect the optimal choice. Revisiting Monotonacity • As all indifference curves are strictly downward sloping, they will all cross a 45 deg line. y d x Revisiting Monotonacity • Monotonacity, Weak and Strong – We will assume strong, so no thick indifference curves! U(d) Weak Monotonacity U(d) d Strong Monotonacity d Establishing Monotonocity • We need to demonstrate that the indifference curve is downward sloping. – Say U0 = U(x, y) – Solve for y = Y(x, U0), making the implicit function, U = U(x, y), explicit. – Calculate dy/dx Example • Say we have U = x2*y – Solve to get: • • • • y = U/x2 dy/dx = -2U/x3 Also, U = x2*y, So dy/dx = -2y/x – For all U and all x > 0, dy/dx < 0 and nonsatiation holds. • However, it may not be possible to solve for Y explicitly (e.g. U = 6y5 – 3xy + 7x3) dy/dx via Implicit Differentiation • We start with an implicit function (identity) defining an indifference curve. To hold when x changes, y must change too. U 0 U x, y(x ) dU 0 dU x , y ( x ) dx dx dU 0 U x , y ( x ) dx x dx 0 U x , y ( x ) dy y U x , y ( x ) dy y dx U x , y ( x ) x dx U x , y ( x ) dy y dx dx U x , y ( x ) x U x , y ( x ) dy dx U (x, y) U x x x U x , y ( x ) U y (x, y) Uy y https://www.khanacademy.org/math/calculus/differentialcalculus/implicit_differentiation/v/implicit-differentiation-1 Example • Start with U = x2*y dy UX dx UY – And U X 2 xy Uy x dy dx 2 2 xy x 2 2y x – So monotonacity holds as -2y/x < 0 for all x,y >0 Intermediate Micro Version • Take the total differential of U = U(x, y) dU U X dx U Y dy 0 U X dx U Y dy U Y dy U X dx dy dx UX UY • This derivation requires dividing by dx, which bothers some people, but everyone teaches it this way (e.g. Chiang and Wainwright, p. 375) Transformations • It sometimes makes life computationally easier to transform a utility function. • Start with utility function that is well behaved (satisfies all the axioms of preference). • We can transform that function with no loss of information so long as the relationship between any bundles A and B is unchanged. Positive Monotonic Transformations • Two functions with identical ordinal properties are called Positive Monotonic Transformations of one another. • Both functions will create identically SHAPED indifference curves (although the utility value associated with each curve will differ). Order preserving transformations • U = U(x,y), original utility function • UT = UT (U(x,y)), transformation function • UT = UT (U(x,y)) is a positive monotonic transformation if UT ‘(U) >0 for all U. U x y 2 U e 2 xy and U and U T T U 1 , 2 dU T dU ln U , dU T 1 U ln x ln y a n d U e , U 0 , th e n U 1 2U dU T 1 T xy 2 0 , th e n U T xy U dU T e U 0 , th e n U T xy dU U xy a n d U T U, dU T 1 0 dU U xy a n d U T 10 U U , 2 dU T dU 10 2 U 0 fo r U 5 Convexity • That is, MRS diminishes as x increase and y decreases. It is about curvature. This Y Not this Y MRS = 5 MRS = 5 MRS = 2 MRS = 2 MRS = 5 MRS = 1 X MRS = 2 X Digression on indifference curves. Indifference curves are often thought of as level curves projected onto the base plane U=U(x, y) U Y This utility function is strictly concave as drawn X Indifference Curves are Level Curves • Level Curves – Are a slice of the utility function at some U = U0 – Even if the utility function is not concave (as drawn above), but only strictly quasi concave, these level curves bound convex sets – Convex sets and level curves • Any line connecting two points on the same level curve lies within the set • So bundles with more balance than two bundles lying on an indifference curve will provide more utility (the utility function will be “above” any line connecting two points on an indifference curve. • Convexity Convexity – DOES NOT IN ANY WAY indicate that the utility function is convex as opposed to concave. – Convex functions and convex sets are two different concepts. • “Strictly” – Strictly quasi–concave utility function means the utility function has no flat spots and it’s level sets are strictly convex – Strictly convex level curves means the indifference curves have no straight line segments – “Strictly” required to ensure just one optimum Convexity of Preferences Implies Indifference Curves Bound Convex Sets This will hold if the utility function is Strictly Quasi Concave U=U(x, y) U Y Utility Function Not Concave, but Strictly Quasi Concave as the level sets bound convex sets Any point on one of these dotted lines (exclusive of end points), provides more utility than the end points X Convexity • Convexity of preferences will hold if: – dy/dx increases along all indifference curves (it gets less negative, closer to zero) – That is, either: • The level sets are strictly convex • The utility function is strictly quasi-concave Convexity (level curves) • dy/dx increases along all indifference curves • We can use the explicit equation for an indifference curve, y=Y(x, U0) and find 2 d y dx 0 2 to demonstrate convexity. • That is, while negative, the slope is getting larger as x increases (closer to zero). U U 0 Alternatively (level curves) • As above, starting with U(x,y)=U0, dy dx M RS U X ( x, y ) , assum ing M R S = U Y ( x, y ) • So convexity if U X ( x, y ) d 2 U ( x , y ) d y Y 0 2 dx dx dy dx Convexity (level curves) • And, that is U X ( x, y ) d 2 2 2 2 U U U U U U U yy U ( x , y ) d y xy x y y xx x Y 0 2 3 dx dx Uy *Note that Uxx and Uyy need not be negative and Uy3>0 • What of: – – – – – Ux > 0, monotonacity, nonsatiation Uy > 0, monotonacity, nonsatiation Uxx, ? Uyy, ? Uxy, ? Diminishing MU vs Diminishing MRS • Both involve the idea of satiation. That is, the more you consume, the less you value added consumption. • DMU: Consumption of other goods irrelevant • DMRS: The value of consuming additional units of one good along an indifference curve falls because you are necessarily consuming less of other goods. Strict Quasi-Convexity (utility function) • Convexity of preferences will hold if the utility function is strictly quasi-concave – A function is strictly quasi-concave if its bordered Hessian 0 f f – is negative definite H 0 fx fx f xx 0 and x y H fx f xx f xy fy f yx f yy 0 fx fy H fx f xx f xy 0 fy f yx f yy Negative Definite (utility function) • So the utility function is strictly quasi-concave if – 1. –UxUx < 0 – 2. 2UxUyUxy-Uy2Uxx -Ux2Uyy > 0 • Related to the level curve result: – Remembering that a convex level set comes from this dx 2 ( 2U xU yU xy U y U xx U x U y y ) 2 2 d y Uy 3 2 0 – We can see that strict convexity of the level set and strict quasi-concavity of the function are related, and each is sufficient to demonstrate that both are true. With all Six Axioms/Assumptions Y A B C U (A ) U (B ) U (C ) A B C U(A) U(B) U(C) X Some Utility Functions and their Properties • Homotheticity of Preferences • Elasticity of Substitution • Functional Forms – CES – Cobb-Douglas – Perfect Substitutes – Perfect Compliments Homothetic Preferences • The MRS depends only on the ratio of goods consumed. • So any MRS that can be reduced so that x and y only appear as the ratio (x/y) or (y/x) are considered homothetic. • Changes in income lead to equal percent changes in consumption (income elasticity = 1 for all goods). Elasticity of Substitution, • What is the % change in the ratio of y*/x* when there is a 1% change in MRSxy? Y y*/x* = slope of MRS at x*, y* = slope of tangent line y* (0,0) x* X Elasticity of Substitution, % ( y * / x*) % (M R S) Change in y/x all different Y % change in MRS from the slope of the original tangent line to each of these is the same y* x* X Elasticity of Substitution, • As an elasticity, it is true that % ( y * / x*) % (M R S) d ( y * / x*) d (M R S) (M R S) ( y * / x*) d ln( y * / x*) d ln(M R S) • And, MRS = Ux/Uy U x x *, y * d U x *, y * y , or y* U x x *, y * x * U y x *, y * y* x* d Evaluated at (x*, y*) y* d ln x * U x x *, y * d ln U x *, y * y And another substitution • And at utility maximizing x*and y*, MRS = px/py, so: % ( y * / x*) % (M R S) % ( y * / x*) % (p x / p y ) • Which means, elasticity of substitution can be defined as either of these: d d p x y* p x* y , or px y * x * py y* d ln x* p d ln x p y • Which has some real economic meaning. It is a measure of the magnitude of the substitution effect. Utility Functions • • • • CES Cobb-Douglas Utility Perfect Substitutes Perfect Compliments CES, Constant Elasticity of Substitution • U A n x n 1 N CES utility: N w here A > 0; n 0; • n 1; ρ< 1; ρ 0; > 0 n 1 Generally, this is simplified U A x y And often to 1 U x y w here A 1, = 1 0< ρ 1 U A x y w here A > 0; ρ 1; ρ 0; > 0 w here A > 0; ρ -1; ρ 0; > 0 • or or U x y w h ere A 1, = 0 < ρ 1 U x y w here A 1 = , 0, 1 CES, Constant Elasticity of Substitution • Start with this form and find Ux: U A x y Ux A x y 1 w h ere A > 0 ; ρ -1 ; ρ 0 ; = 1 1 1 x 1 ( 1) Ux x 1 M u ltip ly b y A x A A Ux x 1 A y 1 A x y ( 1) CES, Constant Elasticity of Substitution • Transform the original utility function U T ransform U : A U A 1 x y 1 x y U S ubstitute A Ux x 1 A 1 in to U x x A U A 1 1 1 1 1 Y ielding 1 A 1 A x y ( 1) CES, Constant Elasticity of Substitution • Simplify Ux x 1 A 1 A U A 1 A nd sim plify Ux x Ux ( 1) A S o, U x U 1 A U x 1 ( 1) U A x 1 1 , and sim ilarly, U y U A y CES: MRS and σ • With Ux and Uy we can define MRS and σ: MRS = Ux Uy d d * y * x px p y px p y * y * x CES: MRS U MRS = Ux Uy 1 A x 1 U A y y 1 x Homothetic! CES: σ d d * y * x px p y px p y * y * x 1 y * x * * * U tility m ax requires x and y such that: 1 y p x 1 S o, at U -m ax: * x py * px py CES: σ • Split it up: d d y * x px p y * y* d * x p x d p y px p y * y * x px p y * y * x S plit this into tw o parts, first deal w ith the derivative portion. d d d d * y * x 1 px 1 p y 1 p 1 x p y * y 1 * 1 x 1 px 1 p y 1 1 p 1 x p y 1 1 , and p 1 x * p x y y * CES: σ • And the other part: px p y * y * x px p y * y * x px p y d d y * x px p y * 1 p x 1 p y y* d * x p x d p y 1 1 p x p y 1 1 1 px p y * y * x 1 , and p 1 x * p x y y * CES, Constant Elasticity of Substitution • Bring the parts back together: y* px 1 d * 1 1 x py = p y* 1 d x * p x y • p 1 x p y 1 1 1 p x p y 1 1 1 Yielding 1 1 1 1 • 1 1 1 1 1 1 p 1 x p y 1 1 1 1 , rem em b er, ρ -1 Means along a CES indifference curve, σ is constant… well named. CES: The Mother of All Utility Functions 1 1 , -1, 0 A s , 0, perfect com plim ents A s 0, 1, C obb-D ouglas A s 1, , perfect substitutes Simpler CES • If we go with this simpler CES: U x y w e g et M RS = Ux U y x y 1 1 w h ere 0, 1 y* px d * p x y R em em ber, p y* d x * p x y * * U tility m ax requires x and y such that: x y 1 y* px S o, at U -m ax: * p x y 1 1 1 px py Simpler CES • Which reduces to p x 1 p y 1 p x py 1 1 1 1 1 1 1 1 1 1 , here, 1, 0 A nd this tim e A s , 0, perfect com plim ents A s 0, 1, C obb-D ouglas A s 1, , perfect substitutes Cobb-Douglas • Cobb-Douglas: U = Axαyβ MRS y x Homothetic • When 0<α<1 and 0<β<1 and α+β=1 – α, share of income spent on x – β, share of income spent on y • To get this, transform: UT = (U)(1/(α+β)) – UT = (xαyβ) 1/(α+β) – UT = (x(α/(α+β))y(β/(α+β))) – (α/α+β) + (β/α+β) = 1 • But how is Cobb-Douglas derived from CES? CES to Cobb-Douglas • First, a digression: ˆ L'H opital's rule: lim a m( ) n( ) lim a m ( ) n ( ) if lim m ( ) 0 and lim n ( ) 0 0 • 0 So now we need to split the Utility function into a ratio of two functions of ρ. CES to Cobb-Douglas • Utility function: U ( ) A x • y 1 w here A > 0; ρ -1; ρ 0; 1 A monotonic transformation. T U = ln U A U = T T U = 1 ln x y 1 ln x y ln x y m ( ) ln x y n( ) CES to Cobb-Douglas • And with this T U = ln x y m ( ) ln x y n( ) • Back to L’Hopital’s Rule. L 'H oˆ p ital's ru le: lim 0 S o : lim 0 ln x m( ) n( ) y lim 0 m ( ) n ( ) d ln x y d lim 0 d d CES to Cobb-Douglas • Derivatives and the limits d ln x y x ln x y ln y d lim = lim 0 0 d x y d lim x x 0 lim 0 lim 0 m ( ) n ( ) m ( ) n ( ) ln x y y ln y ln x ln y = ln x ln y ln x y = ln x y U ( ) S o, lim ln ln x y 0 A B ut w e need lim U ( ), not lim U 0 0 T CES, Constant Elasticity of Substitution • A little rearranging: U ( ) S o, lim l n ln x y 0 A e U ( ) l im ln A 0 Ae Ae • =e U ( ) lim ln A 0 ln x y =Ae U ( ) lim ln 0 A Yielding li m U ( ) = A e ln x y A lim e 0 ln x y U ( ) ln A U ( ) lim A m U ( ) li 0 0 A 0 li m U ( ) A x y 0 • And so long as α+β=1, CES becomes Cobb-Douglas as ρ→0 Perfect Substitutes • Start with a version of CES U x y , and 1 1 , w here, 1, 0 lim U x y 1 MRS = α/β, does not depend on x or y, or y/x No diminishing MRS, not homothetic Perfect Substitutes • Example: If Ozarka Water (12oz) and Dasani Water are (24oz), then U = αO+ βD, β=2α U = αO+ 2αD • MRSOD = ½ • Willing to give up ½ a Dasani for 1 Ozarka D 5 10 O Perfect Compliments • U = min(αx, βy), where α, β >0 – Utility = the smaller of αx or βy – Vertex where αx = βy, or where y/x = α/β • Example: You always eat 3 T of Nutella with 2 slices of bread. N on vert. axis, vertex U = min(3B, 2N) When B = 2 and N = 3, U = 6 When B = 4 and N = 3, U = 6 When B = 2 and N = 6, U = 6 When B = 4 and N = 6, U = 12 N where N/B = 3/2, or N = 3/2 B Bread Neoclassical Behavioral Assertion • Consumers endeavor to maximize U U ( xi ) where U(xi) represents the consumer’s own subjective evaluation of derived from the consumption of goods and services, xi. • Under scarcity, consumers must choose among a limited set of bundles described by the budget constraint px M where pi are the prices of xi goods and services and M is consumer income. i i The Hypothesis • All consumers endeavor to maximize U U ( xi ) subject to the budget constraint pi xi M • So we have beaten Utility to death. • Next week we add in the constraint and solve for the optimal x* and y* Spare: MRS diminishing? • U = x+xy+y o MRS = Ux/Uy = (1+y)/(1+x) o Is 2UxUyUxy-Uy2Uxx -Ux2Uyy > 0 ? o 2(1+y)(1+x)*1-(1+x)2*0-(1+y)2*0 o 2(1+y)(1+x) > 0 • U = x2y2 o MRS = 2xy2/2x2y = y/x o Is 2UxUyUxy-Uy2Uxx -Ux2Uyy>0 ? o 2(2xy2)(2x2y)(4xy)-(4x4y2)(2y2)-(4x2y4)(2x2) > 0 o 32x4y4 - 16x4y4 >0 o 16x4y4 > 0 Diminishing MRS U x x, y dx dy d U xy Uy Ux U x , y U xx dx dx y 2 dx Uy N ote: dy dx dx dy U yy U yx dx dx Ux Uy U x x, y d U x , y y dx Ux Ux U xx U xy U y U x U yx U yy Uy U y 2 Uy 2 U x x, y Ux d U xx U y U xy U x U yx U x U yy U x , y Uy y 2 dx Uy M ultiply by: Uy Uy U x x, y d U x , y U U 2 U U U U U U U U 2 y xx y xy x y yx x y yy x 3 dx Uy U x x, y d U x , y 2 U U U U U 2 U U 2 y xy x y xx y yy x 0 3 dx Uy